Problem

For \(A(x)~=~8~+~500/x\), where \(x\) is a positive whole number of produced items and \(A\) is average cost in dollars per item, find the vertical asymptote and exclusion, state \(\text{both}~\text{algebraic}~\text{one}\text{-}\text{sided}~\text{limits}\), and interpret the feasible side in context.

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What this problem is really about

Separate the algebraic graph from the production story. The denominator identifies the excluded boundary and the two-sided algebraic behavior, but only positive whole-number inputs represent actual item counts. Use the feasible side to explain why dividing a fixed cost among fewer items makes the average cost surge, without pretending fractional or negative production is allowed.

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Algebra: denominator zero at x=0, but no vertical asymptote is recorded; x=0 is excluded. Limits: both one-sided limits are reported as finite 8. Context: positive whole-number production is feasible and small positive production is claimed to approach 8 dollars per item.

Algebra: denominator zero and vertical asymptote at x=0; x≠0. Limits: x→0− gives A→+∞, and x→0+ gives A→+∞. Context: only positive whole-number production is feasible; x=0 is impossible, and small positive production makes fixed cost per item very large.

Algebra: denominator zero and vertical asymptote at x=0; x≠0. Limits: x→0− gives A→−∞, and x→0+ gives A→+∞. Context: only positive whole-number production is feasible; x=0 is impossible, and small positive production makes fixed cost per item very large.

Algebra: denominator zero at x=0, but x=0 is treated as an attainable model value rather than an exclusion. Limits: x→0− gives A→−∞, and x→0+ gives A→+∞. Context: zero production is claimed feasible and assigned an unbounded average cost.

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